# Lindeberg-Feller theorem

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A theorem that establishes necessary and sufficient conditions for the asymptotic normality of the distribution function of sums of independent random variables that have finite variances. Let $X _ {1} , X _ {2} \dots$ be a sequence of independent random variables with means $a _ {1} , a _ {2} \dots$ and finite variances $\sigma _ {1} ^ {2} , \sigma _ {2} ^ {2} \dots$ not all of which are zero. Let

$$B _ {n} ^ {2} = \sum_{j=1}^ { n } \sigma _ {j} ^ {2} ,\ \ V _ {j} ( x) = {\mathsf P} \{ x _ {j} < x \} .$$

In order that

$$B _ {n} ^ {-2} \max _ {1 \leq j \leq n } \ \sigma _ {j} ^ {2} \rightarrow 0$$

and

$${\mathsf P} \left \{ B _ {n} ^ {-1} \sum_{j=1}^ { n } ( X _ {i} - a _ {j} ) < x \right \} \rightarrow \ \frac{1}{\sqrt {2 \pi }} \int\limits _ {- \infty } ^ { x } e ^ {- t ^ {2} /2 } d t$$

for any $x$ as $n \rightarrow \infty$, it is necessary and sufficient that the following condition (the Lindeberg condition) is satisfied:

$$B _ {n} ^ {-2} \sum_{j=1}^ { n } \int\limits _ {| x - a _ {j} | \geq \epsilon B _ {n} } ( x - a _ {j} ) ^ {2} d V _ {j} ( x) \rightarrow 0$$

as $n \rightarrow \infty$ for any $\epsilon > 0$. Sufficiency was proved by J.W. Lindeberg [1] and necessity by W. Feller [2].

#### References

 [1] J.W. Lindeberg, "Eine neue Herleitung des Exponentialgesetzes in der Wahrscheinlichkeitsrechnung" Math. Z. , 15 (1922) pp. 211–225 [2] W. Feller, "Ueber den zentralen Grenzwertsatz der Wahrscheinlichkeitsrechnung" Math. Z. , 40 (1935) pp. 521–559 [3] M. Loève, "Probability theory" , Springer (1977) [4] V.V. Petrov, "Sums of independent random variables" , Springer (1975) (Translated from Russian)
How to Cite This Entry:
Lindeberg–Feller theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lindeberg%E2%80%93Feller_theorem&oldid=22748